In this paper we prove the following result: there exists an infinite arithmetic progression of positive odd numbers such that for any term k of the sequence and any nonnegative integer n, each of the 16 integers k−2n, k−2−2n, k−4−2n, k−6−2n, k−8−2n, k−10−2n, k−12−2n, k−14−2n, k2n−1, (k−2)2n−1, (k−4)2n−1, (k−6)2n−1, (k−8)2n−1, (k−10)2n−1, (k−12)2n−1 and (k−14)2n−1 has at least two distinct odd prime factors; in particular, for each term k, none of the eight integers k, k−2, k−4, k−6, k−8, k−10, k−12 or k−14 can be expressed as a sum of two prime powers.

Erdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.